Minus' Solution to

Crash Course?
Posted March 31 - April 6, 1997

Each has to travel 100 meters to get to the center. Adam travels 10N meters in N minutes, so Adam takes 10 minutes to get to the center (which is 100 meters).

For Cathy, when you are adding whole numbers from 1 to N, the sum is obtained by the following formula:

We can use whole numbers to give us an approximation in this case. If N=13, then 13(14)/2 = 91. We know it takes longer than 13 minutes, but in the first 13 minutes, Cathy travels 91 meters.

During the 14th minute, she is travelling at 14 meters/minute. It takes 9/14 minutes to go the last 9 meters, which is approximately 39 seconds, so Cathy takes about 13 minutes, 39 seconds to go the entire distance to the center.

If it takes Cathy approximately 219 seconds longer than Adam to reach the center, we can rest assured that the two won't crash into each other.

Your Solutions

Mr. T and his Little Einsteins never seem to let Minus down. Here is their wonderfully thorough explanation. Hey, Big Tex, have you thought about entering your Little Einsteins in a math competition? They definately have a knack for complete solutions. BTW: Thanks for turning off the CAPS key! Minus' fins were getting tired...

Obviously, Adam will take exactly 10 minutes to get to the center of the square since he is skating at a constant rate of 10 meters/minute. According to Freddie G., you can find the sum of an arithmetic series by multiplying the sum of the 1st and last numbers by the amount of numbers in the series and dividing that product by 2. In this case, since we are starting at 1, we know the product is 100, thus, we can formulate this simple equation to help us estimate the number of minutes for Cathy:

n(n+1)/2 = 100

After algebraically solving for n, we get n=13.65 & some change. However, that "change" is not entirely accurate because that is based on a parabolic increase in speed when in fact at the beginning of each minute, the rate increases all at once, in a stepping-stone fashion. This subtle difference will alter the exact number of seconds between 13 and 14 minutes that Cathy will need for her journey.

That value will be easier calculated as follows:

Since after 13 minutes, the distance traveled is 13(13+1)/2=91m, then Cathy still has 9 meters to go. She will be traveling this 9 meters at a constant rate of 14 meters/minute or 14m/60 sec.

Using a simple proportion, 14/60 = 9/x, we find that the exact number of seconds needed to finish the 100 meter trek is 38 4/7 seconds, which rounds up to 39 seconds.

Therefore, Adam will arrive at the middle of the pond 1st, whuppin up on Cathy by 3 minutes, 39 seconds (rounded to the nearest second).

Now, you may be saying," Hey, back up Big Tex; in your quadratic above you also got 39 seconds, rounded to the nearest second. Was all this other hoopla necessary?" Answer: You betcha! Had the race been only 1 meter shorter, for example, our equation would have still given us 39 seconds , after rounding, whereas the correct answer, to the nearest second, would in fact be 38. This is not because there is flaw in the algebraic armor; on the contrary, our crash would have been caused entirely by pilot error.

Algebra is always RIGHT; sometimes "we" just insist on turning LEFT!!

Minus' friends from "down-under" really know how to entertain him. Connie Tsang and Emma Rohrlach of Methodist Ladies' College in Claremont, Western Australia, used a couple of tables to plot the distance and time of each skater--a good way to solve the problem without using algebra. And, yes, Minus agrees that in real life, Cathy would have beat Adam.

G'day Minus! Hope you are not too hungry, but this will help to soothe your hunger pangs. Following are the two tables of results for the two skaters. It shows how many metres progressed for every minute. With Cathy's results we had to find out how many minutes it took for 100 metres, which was half of the total 200 metres, then times that by 60 to get the seconds as you asked. There are more seconds than minutes so it must fill you up more!!! : )

Metres			Minutes
1			1	
3			2
6			3		
10			4
15			5
21			6
28			7
36			8
45			9
55			10
66			11
78			12
91			13
100			13.643	(819 seconds rounded)
105			14


Metres		Minutes
10			1
20			2
30			3
40			4
50			5
60			6
70			7
80			8
90			9
100			10		(600 seconds)
In total 219 seconds difference. This answer is rounded. Adam won which is totally unfair and sexist and we are sure that in real life the female would win!!!

Pamela Parker, of Robert Smalls Middle School in Beaufort, SC, delighted Minus with her precise attention to detail. She provided our mascot with the answer in seconds as he had requested.

Each distance is 200 meters , but they intersect at the half way point. Therefore, they each have 100 meters to skate. Since Adam is traveling at a steady 10 meters per minute, he will get to the intersecting point in 10 minutes because 10 x 10 = 100. Ten minutes equal 600 seconds.

Cathy takes a different approach on her journey to the intersecting point by skating a meter the first minute, 2 meters the second minute, 3 meters the third minute and so on until she reaches the intersecting point which is at 13 minutes and 39 seconds. To find this, add the meters Cathy travel each minute until you reach 91. Stop there because in the 14th minute of her journey she would have traveled 105 minutes which is past the middle. You find out how many seconds it would take her to travel 9 meters in the 14th minute of the journey. That would take about 38.571427 (about 39) seconds. The total time is 13 minutes 39 seconds which equals 819 seconds. So Adam is the winner by 219 seconds.

Candice Schalit and Taryn Chua, also of Methodist Ladies' College, were not to be out-done by their classmates. Here is their tasty treat for Minus, and a little advice for Cathy:

If Cathy goes from 1m/min, to 2m/min, to 3m/min, to 4m/min, then this is how far she would get:
1min = 1m
2min = 3m
3min = 6m
4min = 10m
5min = 15m
6min = 21m
Each time, we can see that the number of metres added on is number of minutes passed. Meanwhile, Adam is speedrocking at 10m/min. This would mean that to get to the mid-point of his journey (100m, as the total length is 200m) he would have to travel 100m. This would take him 10 min as he is going at the steady speed of 0m/min(unless he manages to crash along his 'crash course'). So he would take 10 min to reach the mid point.

If we work out how far Cathy has progressed in 10 min, then we have to add 10 to 9 to 8 to 7..... This would get her - nowhere, compared to her speedy opponent - 55m in 10min. So Adam crosses the center first.

To find out by how much time, we have to find out how long Cathy takes to reach the center. So if we continue her course:

9min = 45m
10min = 55m
11min = 66m
12min = 78m
13min = 91m
So, in 13min, she has progressed 91m. She is now travelling at 14m/min. If we change this to 9m, we get 9/14 min. So she takes 13 min and 9/14 min to reach 100m. This is exactly 3min and 9/14min(or 39secs) more than Adam.

ps. Cathy should think about giving up skating.

Kaho Chan, from the Math/Computer Science, Teacher Ed. Program at the University of Nevada, Reno, applied a little geometry to his solution. Hats off to the chef for his ingenuity!

From geometry, the diagonals of a square bisect each other into four segments of equal length. We shall call the intersection of AB and CD, point E. AE = CE = 100 meters because AB = CD = 200 meters.

Adam skates at 10 meters/minute, which means that it'll take him 10 minutes to reach point E (because distance = rate * time; 100 meters = 10 meters/minute * 10 minutes).

Cathy has only traveled 55 meters, in 10 minutes, (because she traveled 1 meter in the 1st minute + 2 meters (2nd minute) + 3 meters (3rd minute) + 4m (4th min.) + 5m (5th min.) + 6m (6th min.) + 7m (7th min.) + 8m (8th min.) + 9m (9th min.) + 10m (10th min.) ) towards point E from point C. This leaves her 45 meters to reach point E. So at the 11th minute, she is travelling at 11m/min. which leaves 34 meters. At the 12th minute, she is traveling at 12m/min. which leaves 22 meters. At the 13th minute, she is traveling at 13m/min. which leaves 9 meters. At the beginning of the 14th minute, she instantaneously changes to 14m/min.

Using distance = rate * time, 9m = 14m/min. * time. Solving for time = (9/14) min. = .642857142857 min = 38.6 seconds. Three minutes plus 39 seconds...

Ans.: Adam crossed the center 219 seconds ahead of Cathy.

Oh, Minus was such a pig this week...
and he's got all of you to thank for it!
Mucho gracias!